Basic probability rules
P(A) is always between 0 (impossible) and 1 (certain). For equally likely outcomes, P(A) = (number of favorable outcomes) ÷ (total number of outcomes).
| Rule | Formula |
|---|---|
| Basic probability | P(A) = favorable outcomes ÷ total outcomes |
| Complement | P(not A) = 1 − P(A) |
| Addition (general) | P(A or B) = P(A) + P(B) − P(A and B) |
| Addition (mutually exclusive) | P(A or B) = P(A) + P(B) |
| Multiplication (general) | P(A and B) = P(A) × P(B | A) |
| Multiplication (independent) | P(A and B) = P(A) × P(B) |
| Conditional probability | P(A | B) = P(A and B) ÷ P(B), P(B) > 0 |
| Bayes' theorem | P(A | B) = P(B | A) × P(A) ÷ P(B) |
| Total probability | P(B) = P(B | A)P(A) + P(B | not A)P(not A) |
| Permutations | nPr = n! ÷ (n − r)! |
| Combinations | nCr = n! ÷ (r!(n − r)!) |
Sources: OpenStax: Introductory Statistics 2e §3.1 Terminology; OpenStax: Introductory Statistics 2e §3.3 Two Basic Rules of Probability
Binomial distribution
For n independent trials, each succeeding with probability p, the number of successes X has P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ. Its mean is np and its variance np(1 − p).
Example: the probabilities of k heads in 10 fair coin tosses, computed exactly:
| k | C(10, k) | P(X = k) |
|---|---|---|
| 0 | 1 | 0.000977 |
| 1 | 10 | 0.009766 |
| 2 | 45 | 0.043945 |
| 3 | 120 | 0.117188 |
| 4 | 210 | 0.205078 |
| 5 | 252 | 0.246094 |
| 6 | 210 | 0.205078 |
| 7 | 120 | 0.117188 |
| 8 | 45 | 0.043945 |
| 9 | 10 | 0.009766 |
| 10 | 1 | 0.000977 |
Sources: NIST/SEMATECH: e-Handbook of Statistical Methods §1.3.6.6.18 Binomial Distribution
Normal distribution
The normal density with mean μ and standard deviation σ is f(x) = (1 ÷ (σ√(2π))) × e^(−(x − μ)² ÷ (2σ²)). Standardize with z = (x − μ) ÷ σ to use the standard normal (μ = 0, σ = 1).
About 68.27% of values lie within 1σ of the mean, 95.45% within 2σ and 99.73% within 3σ (the 68–95–99.7 rule).
| z | Φ(z) | P(−z ≤ Z ≤ z) | P(Z > z) |
|---|---|---|---|
| 0 | 0.5000 | 0.0000 | 0.5000 |
| 0.5 | 0.6915 | 0.3829 | 0.3085 |
| 1 | 0.8413 | 0.6827 | 0.1587 |
| 1.28 | 0.8997 | 0.7995 | 0.1003 |
| 1.645 | 0.9500 | 0.9000 | 0.0500 |
| 1.96 | 0.9750 | 0.9500 | 0.0250 |
| 2 | 0.9772 | 0.9545 | 0.0228 |
| 2.326 | 0.9900 | 0.9800 | 0.0100 |
| 2.576 | 0.9950 | 0.9900 | 0.0050 |
| 3 | 0.9987 | 0.9973 | 0.0013 |
Sources: NIST/SEMATECH: e-Handbook of Statistical Methods §1.3.6.6.1 Normal Distribution; OpenStax: Introductory Statistics 2e §6.1 The Standard Normal Distribution